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Conformal uniformization and packings

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Abstract

A new short proof is given for Brandt and Harrington’s theorem about conformal uniformizations of planar finitely connected domains as domains with boundary components of specified shapes. This method of proof generalizes to periodic domains.

Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of specified shapes and with specified combinatorics. The shapes can be arbitrary smooth disks specified up to homothety, for example. The combinatorics of the packing is described by the contacts graph, which can be specified to be any finite planar graph whose vertices correspond to the shapes. This is in the spirit of Koebe’s proof of the Circle Packing Theorem as a consequence of his uniformization by circle domains.

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Authors and Affiliations

  1. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Oded Schramm

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  1. Oded Schramm
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Correspondence to Oded Schramm.

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The author thankfully acknowledges support of NSF grant DMS-9112150.

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Schramm, O. Conformal uniformization and packings. Israel J. Math. 93, 399–428 (1996). https://doi.org/10.1007/BF02761115

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